Perfect Prismatoids are Lattice Delaunay Polytopes
https://doi.org/10.18255/1818-1015-2014-4-47-53
Abstract
A perfect prismatoid is a convex polytope P such that for every its facet F there exists a supporting hyperplane α k F such that any vertex of P belongs to either F or α. Perfect prismatoids concern with Kalai conjecture, that any centrally symmetric dpolytope P has at least 3d non-empty faces and any polytope with exactly 3d non-empty faces is a Hanner polytope. Any Hanner polytope is a perfect prismatoid (but not vice versa). A 0/1-polytope is a convex hull of some vertices of the d-dimensional unit cube. We prove that every perfect prismatoid is affinely equivalent to some 0/1-polytope of the same dimension. (And therefore every perfect prismatoid is a lattice polytope.) Let Λ be a lattice in Rd and D be a polytope inscribed in a sphere B. Denote a boundary of B by ∂B and an interior of B by int B. The polytope D is a lattice Delaunay polytope if Λ∩int B = ∅ and D is a convex hull of Λ∩∂B. We prove that every perfect prismatoid is affinely equivalent to some lattice Delaunay polytope.
About the Authors
M. A. Kozachok
Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation
Russian Federation
аспирант, Gubkina str. 8, Moscow, 119991, Russia
A. N. Magazinov
Steklov Mathematical Institute of Russian Academy of Sciences
Russian Federation
аспирант, Gubkina str. 8, Moscow, 119991, Russia
Russian Federation
аспирант, Gubkina str. 8, Moscow, 119991, Russia
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Review
For citations:
Kozachok M.A., Magazinov A.N. Perfect Prismatoids are Lattice Delaunay Polytopes. Modeling and Analysis of Information Systems. 2014;21(4):47-53. (In Russ.) https://doi.org/10.18255/1818-1015-2014-4-47-53
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